Problem 46 - University Physics with Modern Physics

Question

A glass plate 2.50 \(\mathrm{mm}\) thick, with an index of refraction of 1.40 , is placed between a point source of light with wavelength 540 \(\mathrm{nm}\) (in vacuum) and a screen. The distance from source to screen is 1.80 \(\mathrm{cm} .\) How many wavelengths are there between the source and the screen?

Step-by-Step Solution

Answer

There are approximately 32 wavelengths between the source and the screen.

Step 1: Determine the number of wavelengths in vacuum

First, calculate how many wavelengths fit into the total distance between the source and the screen in vacuum. The total distance is 1.80 cm = 18.0 mm. The vacuum wavelength is given as 540 nm, which is equal to 0.54 mm. Thus, the number of full wavelengths that fit in the distance is calculated by dividing the total distance by the vacuum wavelength: \[ N_{vacuum} = \frac{18.0\text{ mm}}{0.54\text{ mm/wavelength}} = 33.33 \] Thus, in vacuum, 33.33 wavelengths fit.

Step 2: Calculate the effective distance traveled through glass

The glass plate affects the number of wavelengths because light travels slower in a medium with an index of refraction. To find the effective distance that light travels in the glass, divide the thickness of the glass by its refractive index: \[ d_{glass, effective} = \frac{2.50\text{ mm}}{1.40} = 1.7857\text{ mm} \]

Step 3: Calculate the number of wavelengths in the glass

With the effective distance in the glass now known, calculate how many wavelengths fit in that distance. Using the vacuum wavelength, 0.54 mm, calculate the number of wavelengths in the glass: \[ N_{glass} = \frac{1.7857\text{ mm}}{0.54\text{ mm/wavelength}} = 3.31 \] Thus, in the glass, 3.31 wavelengths fit.

Step 4: Determine the total number of wavelengths

The total number of wavelengths is the sum of wavelengths in the air and the effective number in the glass. Thus: \[ N_{total} = N_{vacuum} - N_{glass}\,in\,vacuum + N_{glass} = 33.33 - 4.63 + 3.31 = 31.68 \] Round the final answer to the nearest whole number, since the number of wavelengths must be an integer, resulting in 32 wavelengths total.

Key Concepts

Wave Optics

Wave optics is an essential part of physics that deals with the behavior of light as a wave. Unlike ray optics, which considers light as straight lines, wave optics focuses on phenomena like interference, diffraction, and polarization. These occur because light behaves as a wave. Understanding wave optics is crucial to explaining why light bends, creates patterns, or combines in certain ways when encountering various obstacles or openings.

When studying wave optics, consider the wave nature of light as fundamental. In practical terms, phenomena such as the complex behavior of light passing through different materials can only be explained through the wave approach. This understanding helps us analyze and calculate changes in the wavelength, as seen in exercises like the one involving the glass plate.

Key points to remember about wave optics:

• Interference - when light waves overlap, they can cancel each other out or strengthen.
• Diffraction - light waves spread out after passing through an opening.
• Polarization - orientation changes in the wave oscillations.

Refraction

Refraction is the bending of light as it passes from one medium to another. This occurs because light changes speed when it enters a new medium with a different density. Refraction is governed by Snell's Law, which relates the angles of incidence and refraction to the indices of refraction of the two media. The formula for Snell's Law is:\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]

In the context of the exercise, refraction happens when light moves from air (or vacuum) into a glass plate. The change in speed results in a different path and light wavelength within the glass. Light travels slower in denser materials like glass, resulting in bending.

Understanding refraction is crucial because:

• It explains why objects look distorted or shifted when viewed through glass or water.
• The bending changes the light's path, affecting how it interacts with other waves or objects.

Light Wavelength

The wavelength of light is the distance between two consecutive peaks of a wave. It is crucial for determining the color of light and how it interacts with various materials. In the vacuum, light travels at its maximum speed, and its wavelength is longest. In mediums like glass, the speed decreases, and so does the wavelength.

The wavelength is vital for calculations of interference patterns and the number of wavelengths over distances in different media. In this exercise, we started with a wavelength of 540 nm in a vacuum. This value changes when light enters the glass plate.

Important notes about light wavelength:

• Determines the light's color - different wavelengths correspond to different colors.
• Wavelength changes with medium - it is affected by the speed of light in various materials.
• Knowing the wavelength aids in calculations for optics problems and explains many wave phenomena.

Index of Refraction

The index of refraction is a measure of how much a material slows down light. It is denoted as \( n \) and is a ratio of the speed of light in a vacuum to the speed of light in the material. Materials with higher indices of refraction slow down light more significantly than those with lower indices.

In the exercise, the entire calculation of how light behaves in the glass relies on knowing that the glass has an index of refraction of 1.40. The index allows us to determine how many wavelengths fit into a given distance within a material.

Key concepts about the index of refraction:

• Ratio-based - it compares light speed in vacuum versus a material.
• Affects light behavior - it decides how much the light will bend entering a new medium.
• Essential for optical designs - crucial for lenses, glasses, and other devices that manipulate light.

Medium Change Effect

The medium change effect describes how light transitions from one medium to another, affecting its speed, direction, and wavelength. This effect is a result of both refraction and changes in the index of refraction. It plays a crucial role in determining how light behaves in optical systems.

In the exercise, this effect is observed as light moves from air to the glass plate and back. The number of wavelengths fitting into a specific distance changes because the index of refraction impacts speed and path. Calculations are adjusted to account for this when determining the total number of wavelengths between the light source and screen.

Understanding the medium change effect is essential because:

• Explains phenomena in everyday situations - such as why a straw looks bent in a glass of water.
• Allows precise calculations in optics - vital for technology that relies on the precise control of light.
• Reveals how optical instruments like microscopes and telescopes function.